>For Euclidean geometry I don't know anything really good in English, but>there is a reasonable book coauthored by Garrett Birkhoff called "Basic>Geometry" which is out of print but possibly gathering dust on a shelf in a>high school near you. I think it sometimes uses the odious "2-column>proof" format, but it also has some good points (such as basing the>treatment of angles on what can be called a "naive theory of real numbers">instead of the more cumbersome Euclidean development).

I think this "naive theory of real numbers" is aserious pedagogical mistake. It has the advantage forthe author of making the book easy to write (and mostUS high school textbooks use this approach). But highschool geometry students do not have an intuition forreal numbers; in particular, they have no intuition forthe topology of the real line.

So the "naive theory of real numbers" is a way ofkeeping from talking about things that *must be taught*if the student is to develop an adequate intuition.Without a knowledge of the geometry of the real line,tacitly assumed in this approach, the student will notunderstand calculus (though he may be able to performthe usual dog tricks of college calculus courses).

This approach makes a farce of any attempt at rigor.Axioms and theorems are "supplemented" with appeals tononexistent intuition about real numbers. The studentmay well wonder why he can't appeal to his ownintuition about triangles, which he probablyunderstands much better than real numbers.

Indeed, I have found that none of the enteringFreshmen I've talked to can give a coherent explanationof what real numbers are. Yet their intuition for realnumbers is made into the foundation of high schoolgeometry courses.

Of all math and science courses taught in highschool, geometry is taught the worst. Nearly all thefreshman math and science students I've taught arelacking the basic knowledge one would expect from ageometry course. They remember no theorems, except forthe Pythagorean theorem and a few conditions for thecongruence of triangles. They don't know any axiomscheme for geometry, and they can't prove anything.(Not surprising, some students I've taught have told methat in their high schools, their geometry teachersSKIPPED ALL THE PROOFS!) They can't tell a theorem from its converse, and they have no idea how to negate(or otherwise manipulate) statements with universal andexistential quantifiers. Most crippling of all, THEYARE WHOLLY UNABLE TO VISUALIZE ANYTHING IN THREEDIMENSIONS. Behold the result of removing solidgeometry from the curriculum.

(I ran into this the other day in a physics workshopI'm teaching. I was trying to explain vector crossproducts, and the students were BEGGING me for2-dimensional examples. They were mortified when Ifinally got it across that vector cross products areinherently 3-dimensional.)

As usual, incompetent teachers are largely to blame.But don't blame them too much. A few years ago Iundertook a diligent search for a good high schoolgeometry book in English. After much searching, andmuch rejecting of hopelessly inadequate candidates, Iconcluded that such a high school geometry book doesn'texist. There are no introductory high school geometrybooks in English which are pedagogically andmathematically sound, and which in addition cover thematerial which ought to be covered.

This last sentence was intended as a lead-in toa discussion: what constitutes pedagogical soundnessin a high school geometry course, and what ought tobe covered?